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Folding optimal polygons

by David Dureisseix

I have looked at various ways of folding the "optimal" pentagon & hexagon (the largest regular hexagon within a square of paper). Additive constraints are a mathematically exact construction, a finite number of operations (no iterative method) and, of course, a folding sequence as simple as possible.

Hexagon

  1. The corner B comes in B' on the medium vertical line. This allows us to built the intersection F of the fold AE with the diagonal BD. Reverse the model.
  2. Fold D onto F.
  3. Resuming the construction to get the optimal hexagon is also easy.

Pentagon

The goal is now to fold a regular pentagon, as large as possible, within a square of paper. In origami geometry, there exists a lot of techniques to fold an approximate pentagon. Much less are concerned with exact pentagon, and only one about optimal pentagon: R. Morassi, The elusive pentagon, in the proceedings of the First International Meeting of Origami Science and Technology, H. Huzita, editor, Ferrara, pp. 27- 37, 1989. The one proposed herein is much simple.
  1. fold AD onto AB where D is the middle of the edge in order to build C. is the golden ratio.
  2. bring C on the horizontal mid-crease.
  3. bisect the complementary angle.
  4. bisect again and mark the diagonal BE.
  5. bisect again. I goes on J.
  6. half way: B goes on J. Unfold.
  7. & 8 complete the stellated pentagon.

 
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